to the best of my knowledge, enhance if you know betterquestion wrote:How do I know which backpacking trails allow overnight camping?
Depends on the area. When unsure, call or google the managing agency such as
National Park
National Forest
Bureau of Land Management
National Parks like Grand Canyon and Saguaro require advanced planning as do some high use areas such as Fossil Creek.
Typically you can camp anywhere in a forest not in a National Park. High use areas have special regulations. Seems one general rule is to camp away from water sources.
Description pages here on HAZ have a permit/fee section.
examples
https://hikearizona.com/decoder.php?ZTN=16 has various links to regulations / permits
https://hikearizona.com/decoder.php?ZTN=116 has a map that shows restriction boundary
Suggested Reading
Intro to Arizona Backpacking / Options